In the number series 6, 26, 96, 276, 650, the next term is _______. – UGC NET Question

The terms follow the rule aₙ = n⁴ + 5n with n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 6, 26, 96, 276 and 650, exactly the given sequence. Applying the same expression for n = 6 produces 1326. Therefore 1326 is the unique value that continues this polynomial pattern without any change in the rule.

Option A:
Option A gives 1326, which is precisely the value obtained from aₙ = n⁴ + 5n when n = 6. It preserves the same functional relationship between n and aₙ that generates all earlier terms. Hence this option correctly represents the next term of the series.

Option B:
Option B proposes 1314, which is close but does not equal the value from n⁴ + 5n at n = 6. To obtain 1314 we would need to subtract 12 from the polynomial result only at this step. Since no such irregular adjustment appears in the previous terms, 1314 cannot be the correct continuation.

Option C:
Option C suggests 1338 as the next term. This number is 12 greater than the value predicted by the rule for n = 6. Introducing such an extra increment only at the sixth term would break the consistent algebraic pattern, so 1338 is not acceptable.

Option D:
Option D gives 1350, which deviates even further from the calculated value 1326. No single formula of the form n⁴ + 5n can generate the first five terms and then jump to 1350 next. Because it would require altering the rule at the last term, this option is incorrect.